Geometric invariant theory for geometers - MathOverflow

Geometric invariant theory for geometers

2012-08-02T14:17:14Z

I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry.

So my question is if there is any nice reference where people explain geometric invariant theory from a geometric viewpoint. In particular, I am looking for a good reference where the analogies between algebraic geometry and differential geometry are pointed out.

Answer by Eugene Lerman for Geometric invariant theory for geometers

2012-08-02T17:19:02Z

I would recommend a look at chapter 8 of the third edition of Geometric invariant theory by Mumford, Forgarty and Kirwan. It describes a connection between GIT and Hamiltonian group actions in symplectic geometry.

(edit) You may also like Moment maps and geometric invariant theory by Chris Woodward.

Answer by Jason Polak for Geometric invariant theory for geometers

2012-08-02T18:43:31Z

If you just want to get a feeling for invariant theory, here are some books that aren't necessarily comprehensive but nevertheless are enlightening at a more leisurely pace as compared to GIT, which would be useful for someone who isn't as familiar with algebraic groups and algebraic geometry:

Santos and Rittatore - Actions and Invariants of Algebraic Groups: Minimal prerequisites. A very gentle introduction to some aspects of invariant theory, including some motivation via Hilbert's 14th problem. This book also contains most of the required theory of linear algebraic groups.
Dolgachev - Lectures on Invariant Theory: This takes a more geometric viewpoint and might be something you are interested in. This only requires some basic knowledge of algebraic geometry.
Schmitt - Geometric Invariant Theory and Decorated Principal Bundles: this might also be interesting if you are interested in the geometric applications and the related geometry, though I haven't looked into this book very much, but Part 1 does contain a fairly leisurely-looking introduction to GIT

There is also Popov's and Vinberg's treatise "Invariant Theory" in the Ecyclopedia of Mathematical Sciences Volume 55 (Springer) which contains a good summary of the classical results in characteristic zero.

Answer by Buschi Sergio for Geometric invariant theory for geometers

2012-08-03T11:02:33Z

For a "more classical" point of view:

"An introduction to Invariants and Moduli " by S. Mukai.

Fogarty J. "Invariant theory" (Benjamin, 1969)

For a introduction to Mumford's:

P. E. Newstead, "Introduction to Moduli Problems and Orbit Spaces"

Anyway you have to learn (before or after) a Gothendieck-categorical background:

Fondements de la géométrie algébrique (Grothendick). The Hilbert schema chapter is very important (need the Hartshorne "Algebraic Geometry" as base)

Or in more gentle way: Fundamental Algebraic Geometry. Grothendieck's FGA Explained - Fantechi B., Göttsche, L., Illusie L.

Answer by kalafat for Geometric invariant theory for geometers

2012-08-06T00:17:56Z

Read the survey in my article and go over the references therein. It is written with exactly similar intentions you have asked for.